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Correlated Matrices.md

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Hermitian

randHermitian

Symmetric

randSymmetric

Elliptic Matrices

randElliptic

RMT: Semicircle Law

The semi-circular law, lecture notes by Terence Tao

using Plots, RandomMatrix, LinearAlgebra, Distributions
N = 2000
colors = [:red,:green,:blue,:purple]
anim = @animate for n = (1:2:50...,51:20:N...,N:-20:51...,50:-2:1...)
    M = randHermitian(Normal(0,1), n,norm=true) 
    histogram(eigvals(M), normed=true, nbins=100,ratio = 7, label="n=$n")
    plot!(x->1/2pi*sqrt(4-x^2), -2:0.001:2, linewidth = 5, label="Semicircle",c=[rand(colors) for _=-2:0.001:2])
    xlims!(-2.5,2.5)
    ylims!(0,0.5)
    title!("Semicircular Law for Random Hermitian Matrices")
end

gif(anim,"Semicircle.gif",fps=20); nothing

RMT : Elliptic Law

THE ELLIPTIC LAW by NGUYEN and O’ROURKE


using Plots, RandomMatrix, LinearAlgebra, Distributions
gr()  
Plots.reset_defaults() 
N = 600

colors = [:red,:green,:blue,:purple]
 
anim = @animate for n = (1:50...,51:10:N...,N:-10:51...,50:1...)
    
randElliptic(500,r=((N-2*n)/N)+0.0001,norm=true)|>eigvals|>x->scatter(x,ylims=(-2,2), xlims=(-2,2),ratio=1,label="ρ = $(round(((N-2*n)/N)+0.0001,digits=2))")

title!("Elliptical Law")

end 
gif(anim, "Elliptic.gif", fps = 10); nothing